From Wikipedia, the free encyclopedia
In commutative algebra, given a homomorphism A → B {\displaystyle A\to B} of commutative rings, B {\displaystyle B} is called an A {\displaystyle A} -algebra of finite type if B {\displaystyle B} can be finitely generated as an A {\displaystyle A} -algebra. It is much stronger for B {\displaystyle B} to be a finite A {\displaystyle A} -algebra, which means that B {\displaystyle B} is finitely generated as an A {\displaystyle A} -module. For example, for any commutative ring A {\displaystyle A} and natural number n {\displaystyle n} , the polynomial ring A [ x 1 , … , x n ] {\displaystyle A[x_{1},\dots ,x_{n}]} is an A {\displaystyle A} -algebra of finite type, but it is not a finite A {\displaystyle A} -algebra unless A {\displaystyle A} = 0 or n {\displaystyle n} = 0. Another example of a finite-type homomorphism that is not finite is C [ t ] → C [ t ] [ x , y ] / ( y 2 − x 3 − t ) {\displaystyle \mathbb {C} [t]\to \mathbb {C} [t][x,y]/(y^{2}-x^{3}-t)} .
This article
needs attention from an expert in Mathematics. See the
talk pagefor details.
WikiProject Mathematics may be able to help recruit an expert. (August 2023)The analogous notion in terms of schemes is that a morphism f : X → Y {\displaystyle f:X\to Y} of schemes is of finite type if Y {\displaystyle Y} has a covering by affine open subschemes V i = Spec ( A i ) {\displaystyle V_{i}=\operatorname {Spec} (A_{i})} such that f − 1 ( V i ) {\displaystyle f^{-1}(V_{i})} has a finite covering by affine open subschemes U i j = Spec ( B i j ) {\displaystyle U_{ij}=\operatorname {Spec} (B_{ij})} of X {\displaystyle X} with B i j {\displaystyle B_{ij}} an A i {\displaystyle A_{i}} -algebra of finite type. One also says that X {\displaystyle X} is of finite type over Y {\displaystyle Y} .
For example, for any natural number n {\displaystyle n} and field k {\displaystyle k} , affine n {\displaystyle n} -space and projective n {\displaystyle n} -space over k {\displaystyle k} are of finite type over k {\displaystyle k} (that is, over Spec ( k ) {\displaystyle \operatorname {Spec} (k)} ), while they are not finite over k {\displaystyle k} unless n {\displaystyle n} = 0. More generally, any quasi-projective scheme over k {\displaystyle k} is of finite type over k {\displaystyle k} .
The Noether normalization lemma says, in geometric terms, that every affine scheme X {\displaystyle X} of finite type over a field k {\displaystyle k} has a finite surjective morphism to affine space A n {\displaystyle \mathbf {A} ^{n}} over k {\displaystyle k} , where n {\displaystyle n} is the dimension of X {\displaystyle X} . Likewise, every projective scheme X {\displaystyle X} over a field has a finite surjective morphism to projective space P n {\displaystyle \mathbf {P} ^{n}} , where n {\displaystyle n} is the dimension of X {\displaystyle X} .
Bosch, Siegfried (2013). Algebraic Geometry and Commutative Algebra. London: Springer. pp. 360–365. ISBN 9781447148289.
RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4